Duality theories of topological groups and topological algebras

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The concept of duality dates back to the time of the French mathematician Evariste Galois (1811-1832), who formulated the criterion for solubility of polynomial equations in a single variable by radicals (n-th roots). At the heart of Galois' solution lies a correspondence between the algebraic structures containing some of the solutions of a given polynomial (called "intermediate field extensions") and substructures of an algebraic structure invented by Galois that is typical to the polynomial (the "Galois group" of the polynomial). An important property of this correspondence is that it reverses the inclusion relation: The larger the intermediate field extension is, the smaller the corresponding substructure is, and vice versa. Although dualities frequently occur in mathematics, they are less often recognized as such. For instance, the starting point of affine algebraic geometry is a duality between sets zero-sets of polynomials in several variables ("algebraic sets") and sets of polynomials with special properties ("radical ideals"). The main feature common to all meaningful dualities is the collection of pairs of dual properties, that is, pairs (P,Q) such that an object satisfies P if and only if its dual possesses Q. The long-term goal of the proposed research is discovering a duality for topological groups with sufficiently many interesting pairs of properties (P,Q) of the above-mentioned type. I do not rule out the possibility that such duality does not exist, in which case, I hope to prove that it is impossible to construct such a duality. In the course of this project, I anticipate to also investigate mathematical objects that are closely related to quantum-mechanics (namely, operator algebras and their generalizations). Any progress in this program outlined will not only contribute to the understanding of topological groups and algebras, but will also considerably stimulate research in these areas, and will cause noticeable "brain drain." From the point of view of the effect on the mathematical community, duality theories often serve as tunnels between two or more otherwise remote areas of mathematics. Thus, dualities have the potential of creating synergy between researchers of distant areas.